On Symmetric Units in Group Algebras

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K . The anti-automorphism g 7→ g of G can be extended linearly to an anti-automorphism a 7→ a∗ of KG . Let S∗(KG) = {x ∈ U(KG) | x∗ = x} be the set of all symmetric units of U(KG) . We consider the following question: for which groups G and commutative rings K it is true that S∗(KG) is a subgroup in U(KG) . We answer this question when either a) G is torsion and K is a commutative G -favourable integral domain of characteristic p ≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.